Self-assembly of anisotropic particles on curved surfaces

TL;DR

This work addresses how surface curvature interacts with anisotropic patchy particle geometry to reshape self-assembly on curved 2D manifolds. Using large-scale molecular dynamics of pentavalent patchy particles confined to sinusoidal surfaces with curvature parameter $\mathcal{C}=d/\lambda$ and fixed amplitude $h/d$, the authors construct a geometric phase diagram as a function of $\mathcal{C}$ and surface coverage $\phi$, revealing curvature-induced square and hexagonal ordering, coexistence regimes, hidden orientational textures, and a transition to glassy dynamics at high curvature. Key findings include curvature-driven emergence and suppression of square order, a robust orientational texturing tied to local curvature, and symmetry-sensitive behavior (square vs triangular substrates) that can frustrate long-range order, all underpinned by both enthalpic and entropic curvature effects on the chemical potential. Collectively, the results establish curvature as a powerful design axis for directing mesoscale morphologies and dynamics in anisotropic assemblies, with potential implications for membrane phenomena and curvature-responsive materials, and they lay groundwork for exploring dynamic or responsive curved surfaces in future work.

Abstract

The surface curvature of membranes, interfaces, and substrates plays a crucial role in shaping the self-assembly of particles adsorbed on these surfaces. However, little is known about the interplay between particle anisotropy and surface curvature and how they couple to alter the free energy landscape of particle assemblies. Using molecular dynamics simulations, we investigate the effect of prescribed curvatures on a quasi-2D assembly of anisotropic patchy particles. By varying curvature and surface coverage, we uncover a rich geometric phase diagram, with curvature inducing ordered structures entirely absent on planar surfaces. Large spatial domains of ordered structures can contain hidden microdomains of orientational textures imprinted by the surface on the assembly. The dynamical landscape is also reshaped by surface curvature, with a glass-like state emerging at modest densities and high curvature. Changes to the symmetry of the surface curvature are found to result in distinct structures, including phases with mesoscale ordering. Our findings show that the coupling between surface curvature and particle geometry opens an unexplored space of morphologies and structures that can be exploited for material design.

Figures (8)

• Figure 1: (a) Schematic representation of a patchy particle. The core is shown in gray while the equatorial and polar patches are shown in purple and green respectively. Patches are enlarged for clarity. (b) Illustration of a subsection of the two dimensional sinusoidal surface on which the patchy particles are confined. The surface is colored by the mean curvature $H$ normalized by its maximum value $H_{\mathrm{max}}$. (c) Side view of the surface displayed in (b).
• Figure 2: Phase Diagram of pentavalent patchy particles confined to a 2D sinusoidal surface as a function of surface coverage $\phi$ and curvature $\mathcal{C}$, with $\varepsilon_{\mathrm{patch}} = 10 k_BT$ and $h/d = 2^{-1/6}$. The green circle morphology marker indicates the isotropic fluid (IF), the red square indicates the square solid (SS), and the blue hexagon indicates the hexagonal (H) phase. States of coexistence are marked by two symbols corresponding to the respective homogeneous phases. Shaded symbols indicate that the states observed may not be the thermodynamic ground state as the the sluggish dynamics preclude us from conclusively determining this. Particles in the representative snapshots are colored by the difference in their two local bond orientation order parameters (details are provided in Methods) $\Delta \psi=|\psi_4| - |\psi_6|$ to distinguish tetratic (green) and hexatic (purple) order from disorder.
• Figure 3: Local density distributions for $\mathcal{C}$ below $(\mathcal{C}= 0.137)$, inside $(0.148 \leq \mathcal{C} \leq 0.190)$ and above $(\mathcal{C}= 0.220)$ the coexistence region at bulk surface coverage $\phi = 0.668$. With increasing $\mathcal{C}$ the distribution starts as uni-modal indicating a homogeneous fluid, but then separates into two peaks at low and high surface coverage indicating phase separation. The symbols represent simulation results while solid lines indicate single or double Gaussian fits to the peaks. The height and position of the peaks are non-monotonic functions of curvature. At extreme curvature, (rightmost panel) the system returns to a disordered fluid. Simulation snapshots are included for each distribution where particles are colored by their tetratic bond orientational order parameter, $|\psi_4|$ (for details see Methods).
• Figure 4: Surface texturing of the square solid. (a) Schematic illustrating the alignment of equatorial patches (in purple) with the surface normal. Polar patches are not shown for clarity. Orientational texturing of square solid-fluid coexistence for (b) $\phi =0.668$ and $\mathcal{C} = 0.144$; (c) $\phi =0.70$ and $\mathcal{C} = 0.126$. (d) Orientational texturing for a pure square solid, $\phi =0.75$ and $\mathcal{C} = 0.120$. For all snapshots, particles are colored by $\cos{\theta^{\mathbf{n}}}$. Anti-aligned particles are red ($\cos{\theta^{\mathbf{n}}} = -1$) and aligned particles are blue ($\cos{\theta^{\mathbf{n}}} = 1$). To assist with visualizing the surface periodicity, we display a square grid (gray) with a spacing of $\lambda$.
• Figure 5: Orientational alignment and its correlation with local curvature. (a) Joint probability distribution of the local mean curvature and the alignment of a particle's equatorial patches with the surface normal, for a pure square solid $\phi=0.75$ and $\mathcal{C}=0.120$. (b) Spatial self-correlation of particle orientations $\left(G_{\theta^{\mathbf{n}}\theta^{\mathbf{n}}}\right)$ and spatial cross-correlation between particle orientation and mean curvature $\left(G_{\theta^{\mathbf{n}}H}\right)$. Inset shows a cosine fit to the periodic undulations of each correlation function. The wavelength of these undulations correspond to $\mathcal{C}=0.120$ for both $G_{\theta^{\mathbf{n}}H}$ and $G_{\theta^{\mathbf{n}}\theta^{\mathbf{n}}}$.